# Properties

 Label 6229.a.6229.1 Conductor 6229 Discriminant 6229 Mordell-Weil group $$\Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -2x^4 + 3x^3 - 3x^2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + 3x^3z^3 - 3x^2z^4$ (dehomogenize, simplify) $y^2 = x^6 - 6x^4 + 14x^3 - 11x^2 + 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, 3, -2], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, 3, -2]), R([1, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 2, -11, 14, -6, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$6229$$ = $$6229$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$6229$$ = $$6229$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-120$$ = $$- 2^{3} \cdot 3 \cdot 5$$ $$I_4$$ = $$11172$$ = $$2^{2} \cdot 3 \cdot 7^{2} \cdot 19$$ $$I_6$$ = $$-507864$$ = $$- 2^{3} \cdot 3 \cdot 7 \cdot 3023$$ $$I_{10}$$ = $$25513984$$ = $$2^{12} \cdot 6229$$ $$J_2$$ = $$-15$$ = $$- 3 \cdot 5$$ $$J_4$$ = $$-107$$ = $$- 107$$ $$J_6$$ = $$389$$ = $$389$$ $$J_8$$ = $$-4321$$ = $$- 29 \cdot 149$$ $$J_{10}$$ = $$6229$$ = $$6229$$ $$g_1$$ = $$-759375/6229$$ $$g_2$$ = $$361125/6229$$ $$g_3$$ = $$87525/6229$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : -1 : 1)$$ $$(1 : -2 : 1)$$
$$(3 : -4 : 1)$$ $$(2 : -17 : 3)$$ $$(5 : -25 : 2)$$ $$(3 : -27 : 1)$$ $$(2 : -36 : 3)$$ $$(5 : -128 : 2)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-36,3],C![2,-17,3],C![3,-27,1],C![3,-4,1],C![5,-128,2],C![5,-25,2]];

Number of rational Weierstrass points: $$0$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

## Mordell-Weil group of the Jacobian

Group structure: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z)^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.303214$$ $$\infty$$
$$(1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.053190$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.015602$$ Real period: $$18.54212$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.289307$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$6229$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 125 T + 6229 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.